Curvature line parametrization from circle patterns

arXiv:0706.3221v1 [math.DG] 21 Jun 2007

Curvature line parametrization from circle patterns A.I. Bobenko and S.P. Tsarev∗ February 1, 2008

Department of Mathematics Technische Universit¨at Berlin Berlin, Germany e-mails: [email protected] [email protected] [email protected]

Abstract We study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by respective discrete nets (circular nets and planar quadrilateral nets) with edges of order ǫ. It is shown that choosing the points of discrete nets on the smooth surface one can obtain the order ǫ2 approximation globally. Also a simple geometric construction for approximate determination of principal directions of smooth surfaces is given.

1

Introduction

Discrete conjugate nets, defined as mappings Z2 −→ R3 with the condition that each elementary quadrangle is flat, and discrete nets of curvature lines, defined as discrete conjugate nets with additional property of circularity of the four vertices of every elementary quadrangle, play an important role in the contemporary discrete differential geometry (see e.g. [2] for a review) and have important applications to computer-aided geometric design [12]. This paper is devoted to a carefull study of the order of approximation of discrete nets to a given smooth conjugate net or the net of curvature lines on a smooth surface that can be achieved. We also present a remarkably simple approximate construction On leave from: Krasnoyarsk State Pedagogical University, Russia. SPT acknowledges partial financial support from the DFG Research Unit 565 ‘Polyhedral Surfaces” (TU-Berlin) and the RFBR grant 06-01-00814. ∗

1

of principal directions on a given smooth surface using the standard primitive of discrete differential geometry: an elementary circular quadrangle inscribed into the surface. Roughly speaking, our results show that the previously known upper bounds ([3]) can be made one order better for conjugate nets and discrete nets of curvature lines. Moreover, one can impose one additional geometric requirement: all vertices of approximating discrete nets should lie on the original smooth surface. The paper is organized as follows. In Section 2 we present a method of construction of principal directions in a neighborhood of a non-umbilic point on a smooth surface using only the four points of intersection of the surface with an infinitesimal circle. Our method is in fact based on a third-order approximation of the given smooth surface and gives an ǫ2 -approximation of principal directions if the infinitesimal circle has the radius ǫ. Different methods for approximation of principal directions have been proposed recently, see for example [8]. Re-meshing techniques based on determination of principal directions are widely used in contemporary computer graphics ([1]). Section 3 is devoted to the study of the order of approximation of a single elementary quad of a discrete conjugate net (resp. discrete circular net) to the respective infinitesimal quadrangle of a the smooth net of conjugate lines (resp. curvature lines) on the original smooth surface. The results of this Section are used in Section 4 where we prove the main global results on the order of approximation of respective discrete nets to a given smooth conjugate net or the net of curvature lines on a smooth surface. We use the following terminology and notations: d(A, S) is the distance from a point A to another point S (or a line, plane, etc.). A point A is said to be ǫk -close to S if d(A, S) = O(ǫk ), that is d(A, S) ≤ Cǫk for ǫ → 0. A is said to lie at ǫk distance from S, if d(A, S) ∼ Cǫk for some C 6= 0, we will denote this hereafter as d(A, S) ≃ ǫk . The same terminology and notations will be used in characterization of the asymptotic behavior for other (numeric) quantities depending on ǫ.

2

Principal curvature directions from circles

In this Section we show that one can use the elementary quadrangle of a discrete circular net inscribed into a smooth surface (see Section 3.2 for more details) for a rather precise construction of the principal directions on a smooth surface. • Euclidean construction. Take three points A, B, C on a smooth surface f : Ω −→ R3 without umbilic points, such that d(A, B) ≃ ǫ, d(B, C) ≃ ǫ and the angle ∠ABC is not ǫ-close to 0 or 2π.Then the circle ω passing through A, B, C has radius of order O(ǫ) as well; from elementary topological considerations one concludes that ω has at least one more point of intersection with f (Ω). Denote it (or one of them) as D. Take the straight lines (AC), (BD) and the point of their intersection Z = (AC) ∩ (BD) (see Fig. 1). The bisectors b1 and b2 of the angles formed at Z by (AC) and (BD) are obviously perpendicular. 2

In fact b1 and b2 approximate the directions of smooth curvature lines on the surface: as we prove below, the directions of b1 and b2 are ǫ-close to the principal directions at any point M ∈ f (Ω) which is ǫ-close to A, B, C, D. In fact one can find the principal directions with only O(ǫ2 )-error at some specific point Zf of f (Ω) which “lies over Z”. Note that the order of the points A, B, C, D on the circle ω is important for finding the principal direction with the error of order ǫ2 . Namely, let Zf be the intersection of f (Ω) with the straight line l passing through Z and perpendicular to the plane πǫ = (ABC). Take the orthogonal (parallel to l) projections p1 and p2 of the bisectors b1 and b2 onto the tangent plane πf of f (Ω) at Zf . Theorem 1 For a general smooth surface f (Ω) in a neighborhood of a non-umbilic point the Euclidean construction given above produces the directions p1 , p2 which are ǫ2 -close to the exact principal directions at Zf . In order to prove this Theorem we establish at first some simple but remarkable facts about planar quadrics and Dupin cyclides (Lemma 2 and Theorem 3) and give another (M¨obius-invariant) construction of the principal directions (Fig. 2). In fact, the intersection curve δ = f (Ω) ∩ πǫ is ǫ2 -close to a quadric — the Dupin indicatrix of f (Ω) at P (see the Appendix); this explains in particular why we may assume that the number of intersection points of ω and f (Ω) is exactly four. If we take then an arbitrary circle ω in a plane, intersecting a quadric in points A, B, C, D (cf. Fig. 1 for the elliptic case; the order of the points A, B, C, D on the circle is not necessarily consecutive for Lemma 2 below to hold), then the directions of the angles formed by the straight lines (AC) and (BD) are parallel to axes of the quadric:

C D Z A

b1

b2 B

Figure 1: Construction of the axes directions of a quadric: A, B, C, D are the four points of intersection with a circle, b1 and b2 are the bisectors of the angles between (AC) and (BD). The bisectors b1 and b2 are parallel to the axes of the quadric

3

Lemma 2 For any plane quadric with four point of intersection with a circle the bisectors b1 and b2 (shown on Fig. 1 for the elliptic case) are parallel to the axes of the quadric. Proof. For simplicity we give the proof only for the case of a non-degenerate central quadric Q(x, y) = (K1 x2 + K2 y 2 ) − 2ǫ2 = 0; the other cases can be proved along the same lines. The coordinates of the intersection points A, B, C, D satisfy the system  Q(x, y) = (K1 x2 + K2 y 2) − 2ǫ2 = 0, (1) ω(x, y) = (x − xǫ )2 + (y − yǫ )2 − rǫ2 = 0. They also lie on the one-parametric sheaf of quadrics Qλ : {Q(x, y) + λω(x, y) = 0}. Obviously the directions of the axes of all Qλ are the same. On the other hand, for some values of λ this quadric is degenerate. These values are the three roots λi of the equation K1 + λ 0 −2λx ǫ = 0. (2) 0 K + λ −2λy 2 ǫ −2λxǫ −2λyǫ 2ǫ2 − λr 2 ǫ

Two of them correspond to the degenerate quadrics composed of the pairs of straight lines {(AB), (CD)} and {(AD), (BC)} shown on Fig. 1. The third degenerate quadric is the pair of straight lines {(AC), (BD)}, shown on Fig. 1. Since the bisectors of these pairs of lines define exactly the axes of the respective degenerate quadric Qλ , we have proved that their directions are the same as for Q0 = {Q(x, y) = 0}. 2 Note that we have not used the ordering of the points A, B, C, D on the circle ω or on the quadric, so the statement of Lemma 2 holds true also for the bisectors of the angles formed by the lines (AB) and (CD), as well as (AD) and (BC). On the other hand in order to obtain the approximation of order ǫ2 below we fix the ordering as shown on Fig. 1. Next we need to reformulate the Euclidean construction given above in terms of M¨obius geometry: We start with a smooth surface f : • M¨ obius-invariant construction. 3 Ω −→ R . Take a sphere S tangent to f (Ω) at some point P , such that for its radius R and the principal curvatures K1 , K2 (K1 < K2 ) of f (Ω) at P the inequalities K1 < 1/R < K2 hold (with non-infinitesimal differences |K1 − 1/R| and |1/R − K2 | ). In this case S intersects an ǫ-patch Ωǫ on f (Ω) along two smooth lines γ1 , γ2 (Fig. 2): Ωf ∩ S = γ1 ∪ γ2 , the angle between γ1 and γ2 is non-zero and non-infinitesimal (this follows from the classical lemma of Morse, cf. for example [10]). Take any circle ω ⊂ S of radius ≃ ǫ with P inside such that the distances between P and the four points of intersection of ω and Ωǫ : ω ∩ γ1 = A ∪ C, ω ∩ γ2 = B ∪ D are of order O(ǫ) as well. Construct two auxiliary circles ω1 , ω2 on S defined by the triples of points {A, P, C} and {B, P, D} respectively (not shown on Fig. 2, ωi are very close to γi ). Then 4

the bisectors b1 , b2 of the angles formed by ω1 , ω2 at P obviously lie in the tangent plane to f (Ω) at P . As we show below, b1 and b2 are ǫ2 -close to the principal directions of f (Ω) at P .

D b2

P A γ1

f (Ω)

C

b1 ω

B γ2

S

Figure 2: Approximate M¨obius-invariant construction of the principal directions. A sphere S is tangent to f (Ω) at P , it intersects f (Ω) along two smooth lines γ1 , γ2 . A circle ω ⊂ S of radius ≃ ǫ with P inside gives four points: ω ∩ γ1 = A ∪ C, ω ∩γ2 = B ∪D, the points A, B, C, D should lie on ǫ-distance from P . The bisectors b1 , b2 of the angles formed by the circles ω1 , ω2 (not shown) passing through the triples of points {A, P, C} and {B, P, D} are ǫ2 -close to the principal directions of f (Ω) at P First we prove that for Dupin cyclides (standard M¨obius primitives, playing the role of the osculating paraboloids of the classical Euclidean differential geometry, see [6, 9]) bi are exactly the principal directions: Theorem 3 For a smooth non-umbilic point P on a Dupin cyclide D, the bisectors b1 and b2 of the M¨obius-invariant construction give the principal directions of D at P. Proof. Since our construction of ωi is M¨obius-invariant, we may transform D into one of the normalized Dupin cyclides: a torus, a circular cone or a circular cylinder. We give below a detailed proof for the case of a torus; the other cases can be easily proved along the same lines. Making, if necessary, another M¨obius transformation, we can reduce the configuration to the following: the torus is given by the equation (R2 + x2 + y 2 + z 2 − r 2 )2 = 4R2 (x2 + y 2),

(3)

the point P has the coordinates (R+r, 0, 0) and the sphere S is given by the equation
2 x + ρ − (R + r) + y 2 + z 2 = ρ2 . (4) 5

After a M¨obius inversion (x, y, z) → (x1 , y1 , z1 ) with the center P :  x−R−r   , x1 =   (x − (R + r))2 + y 2 + z 2     y y1 = , (x − (R + r))2 + y 2 + z 2      z   ,  z1 = (x − (R + r))2 + y 2 + z 2

(5)

the sphere (4) will be transformed into a plane x1 = const = µ. We prove now that the intersection of the image of the torus with any plane x1 = µ is a quadric of the form ay12 + bz12 = κ. In fact, from (5) we have:  
2 x − (R + r) + y 2 + z 2 = (x − R − r)/µ (6)

so

z 2 = (x − R − r)/µ −



(3) may be now reshaped to 

x − (R + r)


2

 + y2 .

(7)

2
x − (R + r) /µ + 2(R + r)x − 2rR = 4R2 (x2 + y 2),

so one can express y 2 in terms of x, R, r, µ. Substituting (6), (7) and the  found .
2 2 2 2 2 2 2 2 expression for y into ay1 + bz1 = (ay + bz ) x − (R + r) + y + z one gets a rational expression containing only x, the constants R, r, µ and a, b. A straightforward calculation shows that for a = r(1 + 2µ(R + r)), b = (1 + 2rµ)(r + R) this expression is a constant κ = −R(1 + 2µR)(1 + 2µr)(1 + 2µr + 2µR)/4. Since the images of the circles ω1 , ω2 defined above are now straight lines passing through the points of intersection of the image of the circle ω and the found quadric on the plane x1 = µ, the statement of this Theorem is now equivalent to the statement of Lemma 2. 2 Theorem 4 For an arbitrary smooth surface f (Ω) in a neighborhood of a nonumbilic point P the M¨obius-invariant construction produces the directions b1 , b2 which are ǫ2 -close to the exact principal directions at P . Proof. First we perform a M¨obius transformation that maps the sphere S and the circle ω into themselves and brings P into the (Euclidean) center of ω on S, i.e. into the point P ′ ∈ S such that the distance from P ′ to all the points of ω are equal. One can easily check that our requirement d(P, A) ≃ ǫ, . . . , d(P, D) ≃ ǫ guarantees that after the transformation these distances will remain of order ≃ ǫ. Inside the circle ω on S and in its ǫ-neighborhood in R3 where f (Ωǫ ) lies, the Jacobian of this transformation is limited so the distances of order O(ǫp ), p ≥ 2 will be transformed 6

into distances of the same order. Since the angles do not change, ǫ2 -close directions at P remain ǫ2 -close at P ′ and vice versa. Introduce now a Cartesian coordinate system such that P ′ is its origin, the coordinate plane (xy) is tangent to S (so to f (Ω) as well) and the x, y-axes are the principal directions of f (Ω) at P ′. The plane where ω lies will be given by the equation z = c1 ǫ2 , the circle ω will be defined as x2 + y 2 = c22 ǫ2 and f (Ω) will be defined in a neighborhood of P ′ by its Taylor expansion z = (K1 x2 + K2 y 2)/2 + (a30 x3 + a21 x2 y + a12 xy 2 + a03 y 3) + o(ǫ3 ).

(8)

So the intersection points P1 = A, P2 = B, P3 = C, P4 = D of ω and f (Ω) will have respectively the coordinates xi = ǫxi0 + ǫ2 xi1 + o(ǫ2 ) with (xi0 , yi0 ) being the solutions of  2 x + y 2 = c22 , (9) (K1 x2 + K2 y 2) = 2c1 , so we see that (x10 , y10 ) = (x20 , −y20 ) = (−x30 , −y30 ) = (−x40 , y40 ). The next terms (xi1 , yi1 ) are found after substitution of (xi , yi) into  2 x + y 2 = c22 , (K1 x2 + K2 y 2)/2 + ǫ(a30 x3 + a21 x2 y + a12 xy 2 + a03 y 3 ) = c1 ,

(10)

(11)

cancellation of the second-order terms using (9) and retaining only third-order terms:  2xi0 xi1 + 2yi0 yi1 = 0, (12) 2 3 K1 xi0 xi1 + K2 yi0 yi1 + (a30 x3i0 + a21 x2i0 yi0 + a12 xi0 yi0 + a03 yi0 ) = 0. From (12) and (10) we easily conclude that (x11 , y11 ) = (x31 , y31 ), (x21 , y21 ) = (x41 , y41 ),

(13)

so the O(ǫ2 )-shifted lines l11 , l21 defined by the pairs of points {(ǫx10 + ǫ2 x11 , ǫy10 + ǫ2 y11 ), (ǫx30 + ǫ2 x31 , ǫy30 + ǫ2 y31 )} and {(ǫx20 + ǫ2 x21 , ǫy20 + ǫ2 y21 ), (ǫx40 + ǫ2 x41 , ǫy40 + ǫ2 y41 )} are parallel to the lines l10 , l20 defined by the pairs {(ǫx10 , ǫy10 ), (ǫx30 , ǫy30 )} and {(ǫx20 , ǫy20 ), (ǫx40 , ǫy40 )}. Thus the bisector directions between l11 , l21 in the plane are the same as for the unperturbed lines l10 , l20 ; so they are obviously parallel to the coordinate axes — the principal directions of f (Ω) at P ′ which is ǫ2 -close to both Z0 = l10 ∩ l20 and Z1 = l11 ∩ l21 . Now we have two pairs of circles in R3 : ω10 , ω20 passing through the triples of points {(ǫx10 , ǫy10 ), P ′, (ǫx30 , ǫy30 )} and {(ǫx20 , ǫy20 ), P ′, (ǫx40 , ǫy40 )} and the pair ω11 , ω21 passing through {(ǫx10 + ǫ2 x11 , ǫy10 + ǫ2 y11 ), P ′, (ǫx30 + ǫ2 x31 , ǫy30 + ǫ2 y31 )} and {(ǫx20 + ǫ2 x21 , ǫy20 + ǫ2 y21 ), P ′, (ǫx40 + ǫ2 x41 , ǫy40 + ǫ2 y41 )}. One easily concludes 7

from (13) that ω10 , ω11 are tangent at P ′, as well as ω20 , ω21 . Thus their bisectors are the same and parallel to the axes; the next O(ǫ3 )-terms in approximations for the intersection points A, B, C, D will give a O(ǫ2 )-change in the directions of b1 , b2 . 2 Remark. From the proof of the previous Theorem we see that in fact the M¨obiusinvariant construction gives the same (with O(ǫ2 )-error) principal directions if we will substitute instead of the original surface f (Ω) its tangent paraboloid at P (after the M¨obius transformation described in the beginning of the proof). Also it should be noted that the O(ǫ2 )-approximation was achieved taking into account approximating paraboloid of third order ; the results obtained in this Section are therefore third order results despite the seemingly second-order construction based on an infinitesimal circle. Proof of Theorem 1. Now we are in a position to prove the statement about O(ǫ2 )-approximation of principal directions at Zf in our Euclidean construction. For this we first perform an auxiliary construction in R3 (Fig. 3): C B Zf

ω1

β2

Z b2

A

β1

b1

ω2

ω

D

Figure 3: Auxiliary construction take the circles ω1 , ω2 defined by the point triples {A, Zf , C} and {B, Zf , D}, then for the sphere Σ where all these points and circles lie we take its tangent plane πΣ at Zf . Next we construct another two circles β1 , β2 as the sections of the sphere Σ by the “bisectoral planes” passing through Zf and the lines b1 , b2 respectively. First we remark that the constructed plane πΣ is ǫ2 -close to the tangent plane πf of f (Ω) at Zf : the circles ω1 , ω2 are approximating the osculating circles of the planar sections of f (Ω) by the planes (A, Zf , C) and (B, Zf , D). As one can check by a direct computation for any planar smooth curve γ, if one takes the circle ω passing through three points lying on ǫ-distances from each other on γ, then the tangents to ω and γ at any of the three pins will be ǫ2 -close. So the tangents to the aforementioned planar sections and the tangents to the respective circles ω1 , ω2 8

at Zf are ǫ2 -close. (Here we use the same terminology of “ǫp -closeness” for pairs of lines or planes, this means that the respective angles are O(ǫp ).) This shows that our sphere Σ is “very close” to the sphere S (which should be tangent to f (Ω) at Zf = P ) used in the M¨obius-invariant construction. We also see that our Euclidean construction is nearly reduced to the M¨obius-invariant construction, with the sphere S substituted by Σ: the orthogonal projections (along the line l = (ZZf ) ⊥ πABCD ) of b1 , b2 onto πΣ are nothing but the tangents to the circles β1 , β2 which are ǫ2 -close to the circles which exactly bisect the angles between the circles ω1 , ω2 at Zf . This follows from the fact, that the angle between πABCD and πf is O(ǫ): as one can easily check by a direct computation, in this situation the (non-infinitesimal) angle between any two lines on one of the planes and the angle between their orthogonal projections on the other plane are ǫ2 -close. Let us now estimate the angles between the tangents to the circles βi (lying in πΣ ) and the principal directions of f (Ω) at Zf (lying in πf , which is ǫ2 -close to πΣ ). For this we use the same technology as in the proof of Theorem 4: first we perform a M¨obius transformation leaving Σ and ω invariant and bringing the point Zf to the center O of ω on Σ, then introduce a Cartesian coordinate system with O as its origin and πΣ being its (xy)-plane. Now the equation (8) of the surface f (Ω) will be modified by small linear terms: z = (K1 x2 + K2 y 2 )/2 + (a30 x3 + a21 x2 y + a12 xy 2 + a03 y 3) + σ1 x + σ2 y + o(ǫ3 ), (14) with σi = O(ǫ2 ). The first terms xi0 , yi0 in the Taylor expansions of the coordinates of the points A, B, C, D will be found from the same system (9) so we have the relations (10). The next terms xi1 , yi1 are found from a modified system of the form (12):  2xi0 xi1 + 2yi0 yi1 = 0, 2 3 K1 xi0 xi1 + K2 yi0 yi1 + (a30 x3i0 + a21 x2i0 yi0 + a12 xi0 yi0 + a03 yi0 ) + σǫ21 xi0 + σǫ22 yi0 = 0. We see that the main conclusion (13) still holds; from this point we just follow the guidelines of the proof of Theorem 4 and show that the bisectors of the angles between the circles ω1 , ω2 coincide (up to a O(ǫ2 )-error) with the directions of the x- and y-axes. On the other hand since we know that σi = O(ǫ2 ), using the standard differential-geometric formulas for calculation of the coefficients of the second fundamental form and the principal directions we conclude that the principal directions of f (Ω) given by the equations (14) are ǫ2 -close to the x- and y-axes at Zf as well (as the directions of the straight lines in R3 ). 2 Remark. From the proof of this Theorem we see that in fact one can assume (with O(ǫ2 )-error) in the Euclidean construction that the tangent plane at Zf is the readily constructible plane πΣ .

9

3

Local results

3.1

Smooth and discrete conjugate quads

Suppose that a smooth surface f (Ω) parametrized locally by some curvilinear net of conjugate lines is given: f : Ω −→ R3 , Ω ⊂ R2 = {(u, v)}. Take some initial point A = f (u0, v0 ) and points B = f (u0 + ǫ, v0 ), C = f (u0, v0 + ǫ) at ǫ-distance on the two conjugate lines of the net on f (Ω) and let D = f (u0 + ǫ, v0 + ǫ) be the fourth point of the curvilinear quad on f (Ω). Using the conjugacy condition fuv = a(u, v)fu + b(u, v)fv

(15)

and its derivatives, one can easily estimate the distance from this fourth point to the plane πABC defined by A, B, C; we formulate this as our next Theorem. We remind that smooth conjugate nets are supposed to be non-degenerate, that is the directions of the curvilinear coordinate lines of such a net are non-asymptotic, so the angle between the tangent vectors fu and fv is everywhere different from zero. Theorem 5 For an arbitrary smooth non-degenerate conjugate net f (Ω), d(D, πABC ) = O(ǫ4 ). Proof. Using the standard Taylor expansions one gets: −→ 2 3 AB = ǫfu + ǫ2 fuu + ǫ3! fuuu + o(ǫ3 ), −→ 2 3 AC = ǫfv + ǫ2 fvv + ǫ3! fvvv + o(ǫ3 ), −−→ −→ −→ 3 AD = AB + AC + ǫ2 fuv + ǫ2 (fuvv + fuuv ) + o(ǫ3 ),

(16)

where all derivatives are taken at the point (u0 , v0 ). Taking into account (15) and its derivatives fuuv = au fu + bu fv + afuu + b(afu + bfv ), fuuv = av fu + bv fv + a(afu + bfv ) + bfvv , −→ −→ − −→ we can compute the triple product ((AB × AC) · AD) which gives the volume of the −→ −→ −−→ parallelepiped spanned by the vectors AB, AC, AD: −→ −→ −−→ −→ −→ ǫ3 −→ −→ ((AB × AC) · AD) = ǫ2 (AB × AC) · fuv + (AB × AC) · (fuvv + fuuv ) + o(ǫ5 ) = 2   i ǫ5 h (fu × fv ) · fuvv + fuuv + (fuu × fv ) · fuv + (fu × fvv ) · fuv + o(ǫ5 ) = 2 i 5h ǫ a(fu × fv ) · fuu + b(fu × fv ) · fvv + a(fuu × fv ) · fu + b(fu × fvv ) · fv + o(ǫ5 ) = o(ǫ5 ). 2 10

−→ −→ Since the area of the base of this parallelepiped |AB × AC| ≃ ǫ2 , we get that its height d = o(ǫ3 ), so at least we may state that d = O(ǫ4 ). In fact, using Taylor expansions in (16) up to order O(ǫ4 ) one can obtain after a lengthy computation that i −→ −→ −−→ ǫ6 h (au −ab)(fuu ×fu ) · fv −(bv −ab)(fvv ×fu ) · fv + o(ǫ6 ) (17) ((AB × AC) · AD) = 12 which proves that for a generic conjugate net we have d ≃ ǫ4 . 2 In fact we can even choose M (the fourth point of an elementary planar quad ABCM) on the plane πABC sufficiently close to D and lying on the given smooth surface f (Ω): Theorem 6 For a given smooth non-degenerate conjugate net f : Ω −→ R and the plane πABC constructed as above one can choose a point M ∈ πABC ∩ f (Ω) such that d(M, D) = O(ǫ3 ). Proof. As we prove in the Appendix, the intersection of the plane πABC and the surface f (Ω) in the ǫ-neighborhood of the points A, B, C, D for sufficiently small ǫ is a curve IABC close to a quadric – the Dupin indicatrix of f (Ω), and the angle between πABC and f (Ω) is ≃ ǫ in all points of IABC . According to Theorem 5, d(D, πABC ) = O(ǫ4 ). From this we can easily see that the distance between D and the closest to D point M on IABC should be O(ǫ4 /ǫ) = O(ǫ3 ). 2

3.2

Curvilinear quads of curvature lines and plane circular quads

For a given smooth surface f (Ω) parametrized by curvature lines one can prove a similar result, if we take the circle ωABC passing through the points A, B, C defined as in Section 3.1. Theorem 7 For arbitrary smooth net of curvature lines in a neighborhood of a nonumbilic point A on f (Ω), the distance between the fourth point M of intersection of the circle ωABC with f (Ω) and the point D has order 3: d(D, M) = O(ǫ3 ). In order to prove this theorem we need to establish a few auxiliary Lemmas. Lemma 8 For arbitrary smooth net of curvature lines in a neighborhood of a nonumbilic point A on f (Ω), d(D, ωABC ) = O(ǫ3 ). Proof. Following an approach proposed in [4], we introduce for each of the points A, B, C, D purely imaginary quaternion A, B, C, D. The point A will be chosen as the origin, i.e. A = 0. The basic quaternions I, J, K are chosen to be tangent to the curvature line directions fu , fv and the normal vector to the surface in the initial point A respectively. In view of future applications in the theory of triply 11

orthogonal coordinate systems we include the given surface f (Ω) into such a system f (u, v, w), f (Ω) being one of the coordinate surfaces f (u, v, w = w0 ). This is always possible (see e.g. [7]): one can take for example the one-parametric family of surfaces parallel to f (Ω) and the other two one-parametric families of developable surfaces defined by the curvature lines of f (Ω) and the normals to f (Ω). Introducing for this 3-orthogonal system the Lam´e coefficients Hi (u, v, w) = ~i = ∂i f /Hi and the |∂i f |, ∂1 = ∂/∂u, ∂2 = ∂/∂v, ∂3 = ∂/∂w, normalized vectors V rotation coefficients βik (u) = ∂i Hk /Hi, i 6= k, βii (u) = 0, we have the following relations ([7]): ∂i Hk = βik Hi , ~k = βkiV~i , ∂i V P ~i = − ~ ∂i V s6=i βsi Vs ,

(18)

∂j βik = βij βjk ,

i 6= j = 6 k, P ∂i βik + ∂k βki + s6=i,k βsi βsk = 0,

~1 (u0 , v0 ) = I, where i, j, k ∈ {1, 2, 3}, i 6= j 6= k. In the initial point A we have V ~2 (u0 , v0 ) = J, V~3 (u0 , v0 ) = K. The Taylor expansions for the other points are now V ~2 using (18): easily obtained after differentiation of fu = H1 V~1 , fv = H2 V B = ǫH1 I + C = ǫH2 I + D=B+C

ǫ2 (∂1 H1 I 2

− β21 H1 J − β31 H1 K) + o(ǫ2 ),

ǫ2 (−β12 H2 J + ∂2 H2 J − β32 H2 K) 2 + ǫ2 (β21 H2 I − β12 H1 J) + o(ǫ2 ).

+ o(ǫ2 ),

(19)

Calculating now the quaternionic cross-ratio ([4]) Q = (A−B)(B−C)−1 (C−D)(D− A)−1 and taking its imaginary part, one can see that ImQ = o(ǫ). According to [4] we conclude that ρ = o(ǫ2 ) for the distance ρ from the point D to the circle ωABC of size ≃ ǫ defined by the points A, B, C. 2 In the following we fix an ǫ-patch f (Ωǫ ) of size ≃ ǫ on the surface f (Ω), such that all the points A, B, C, D belong to it. All subsequent constructions will be applied only to this ǫ-patch f (Ωǫ ). From Lemma 16 (proved in the Appendix) we know that the intersection curve IABC = f (Ω) ∩ πABC is ǫ2 -close to the Dupin indicatrix of f (Ω) at some point M. In fact we can take the indicatrix at the following point P instead: choose the “center point” P on f (Ω) such that P is the intersection of f (Ω) and the normal to the plane πABC passing through the center of the circle ωABC . In the Cartesian coordinate system (similar to the system used in the proof of Lemma 16), where the osculating paraboloid at P is P = {z = (K1 x2 + K2 y 2 )/2}, the points A, B, C, D will have

12

coordinates

  2 2 2 ǫ + o(ǫ ) , A − 2ǫ + o(ǫ), − 2ǫ + o(ǫ), K1 +K 8   2 2 ǫ + o(ǫ2 ) , B 2ǫ + o(ǫ), − 2ǫ + o(ǫ), K1 +K 8   2 2 2 C − 2ǫ + o(ǫ), 2ǫ + o(ǫ), K1 +K ǫ + o(ǫ ) , 8   2 2 ǫ + o(ǫ2 ) , D 2ǫ + o(ǫ), 2ǫ + o(ǫ), K1 +K 8

(20)

so for the angle θ between the plane πABC and the tangent plane πP to f (Ωǫ ) at P we have θ = O(ǫ2 ). Moreover, the following is true: Lemma 9 The centers of the Dupin indicatrix QP = P ∩ πABC and ωABC are ǫ2 close. The angles of intersection of ωABC and IABC are ≃ ǫ0 = 1. Proof. From (20) we easily deduce the first statement. The radius of ωABC is √ 2 2 + o(ǫ) and K1K+K + o(ǫ). Since we ǫ 2 + o(ǫ), while the axes of QP are K1K+K 1 2 assume |K2 − K1 | ≻ ǫ on Ω we see that the intersection angles of QP and ωABC , so also of ωABC and IABC , are ≃ ǫ0 = 1. 2 Now using the same technique as in the proof of Lemma 16, we deduce that the Dupin indicatrix at P and IABC are ǫ2 -close. Proof of Theorem 7. We know (Theorem 5) that D is ǫ4 -close to πABC . Using the fact that the angle of intersection of πABC and f (Ωǫ ) is ≃ ǫ (so ≻ ǫ2 ) we see that D is ǫ3 -close to IABC = πABC ∩ f (Ω). On the other hand (Lemma 8) D is ǫ3 -close to the circle ωABC . Since the angles of intersection of ωABC and IABC are ≃ ǫ0 = 1, we see that D and M = IABC ∩ ωABC are ǫ3 -close as well. 2

4 4.1

Global results Conjugate nets

Using the results of Section 3.1 one can try to construct inductively an approximating discrete conjugate net f ǫ for sufficiently small ǫ > 0. At the first glance the following simplest strategy may be applied: starting from the initial point A ≡ f00 = f (u0 , v0 ) on a finite piece of a smooth surface f : Ω −→ R3 parametrized by conjugate coordinate lines, fix two series of points fi,0 = f (u0 + iǫ, v0 ), f0,i = f (u0 , v0 + iǫ) at ǫ-distances on two curvilinear coordinate lines on f (Ω) passing through A. We can ǫ construct the next point f11 as the orthogonal projection of f11 = f (u0 +ǫ, v0 +ǫ) onto ǫ the plane passing through f00 , f10 , f01 ; then f21 as the orthogonal projection of f21 = ǫ f (u0 + 2ǫ, v0 + ǫ) onto the plane passing through f10 , f20 and f11 , etc. This inductive process is shown on Figure 4 with initial points fij = f (u0 + iǫ, v0 + jǫ) on the surface f (Ω) with smooth coordinate lines (shown as dash-lines). The approximating discrete conjugate net consists of the points fijǫ , fi0ǫ ≡ fi0 , f0iǫ ≡ f0i . This “geometric analogue” of the analytic approximation results of [3, 11] should give us, if the results of [3, 11] were directly applicable, an estimate d(fijǫ , fij ) ≤ C(i+ 13

ǫ f13

ǫ f23 ǫ f33

f03 =

ǫ f03

f13

f02 =

ǫ f32

f12

ǫ f00 = f00

f22

ǫ f21

ǫ f11 ǫ f01 = f01

f33

ǫ f22

ǫ f12 ǫ f02

f23

f11

ǫ f10 = f10

f32

ǫ f31

f21

f31

ǫ f20 = f20 ǫ f30 = f30

Figure 4: A discrete conjugate net fijǫ and a smooth conjugate net fij which are ǫ2 -close. The points of fijǫ and fij on the initial curves coincide j)·ǫ4 , so for the complete simple piece f (Ω) we would have d(fijǫ , fij ) = O(ǫ3 ). In fact d(fijǫ , fij ) shows quadratic behavior: d(fijǫ , fij ) ≤ Ci·j·ǫ4 , so on f (Ω) we have a much weaker estimate d(fijǫ , fij ) = O(ǫ2 ). Indeed, a straightforward calculation shows that for the distances d(fijǫ , fij ) of the four vertices of any elementary discrete quadrangle ǫ ǫ ǫ one has d(fi+1,j+1 , fi+1,j+1 ) ≃ d(fi+1,j , fi+1,j ) + d(fi,j+1 , fi,j+1) − d(fijǫ , fij ) + ǫ4 φ(u0 + iǫ, vo + jǫ) where φ(u, v) = (au − ab)(fuu × fu ) · fv − (bv − ab)(fvv × fu ) · fv is the coefficient of ǫ6 in the right hand side of (17). Then for example for special surfaces which have constant φ(u, v) ≡ φ0 6= 0 one easily obtains d(fijǫ , fij ) ≃ i · j · ǫ4 . Paradoxically, a better strategy consists in choosing the points fijǫ of the approximating discrete net on the given smooth surface f , although Theorem 6 suggests that one has worse local error ≃ ǫ3 than the local error ≃ ǫ4 of Theorem 5. The construction we propose below is applicable to arbitrary simple pieces f (Ω) without parabolic points, i.e. points where along with the diagonal coefficient (fuv × fu ) · fv of the second fundamental form (cf. (15)) at least one of the other coefficients (fuu × fu ) · fv , (fvv × fu ) · fv also vanishes. ǫ First we fix the vertices fi0ǫ = fi0 = f (u0 + iǫ, v0 ) and all even “columns” f2i,j = f2i,j = f (u0 + 2iǫ, N v0 + jǫ) in the new approximating discrete conjugate net (marked on Figure 5 as -points). Then we proceed as follows. Two planes defined by the triples of points {f00 , f10 , f01 }, ǫ {f10 , f20 , f21 } intersect along a straight line passing through f10 = f10 . As we show ǫ below in Lemma 10, this straight line intersects f (Ω) in another point f11 which ǫ 3 is close to f11 , d(f11 , f11 ) = O(ǫ ), whereas the distance between this new point ǫ f11 and the line v = v0 + ǫ of the smooth conjugate net is of order O(ǫ4 ). Other 14

f12

ǫ f02 = f02 ǫ f12

ǫ f32

f11

ǫ f01 = f01

ǫ f10 = f10

f31

ǫ f21 = f21 ǫ f31

ǫ f11

ǫ f00 = f00

f32

ǫ f22 = f22

ǫ f20 = f20

ǫ f30 = f30

ǫ f40 = f40

Figure 5: A discrete conjugate net fijǫ with its vertices on f (Ω), which are ǫ2 close to the vertices of a smooth conjugate net fij . The original smooth net fij is schematically shown as a rectangular plane net ǫ ǫ points f31 , f51 , . . . of the first row are constructed in the same way and obviously ǫ , i = 0, 1, . . . of the vertices have the same error estimates. The second row f2i+1,2 of the approximating discrete net is constructed analogously using the fixed points ǫ f2k,2 , k = 0, 1, . . . of the second row on f (Ω) and the constructed complete set of the points fi1ǫ of the first row (cf. Figure 5). ǫ As Lemma 11 below shows, f2i+1,2 are now ǫ5 -close to the smooth line v = v0 +2ǫ ǫ whereas the distance between f2i+1,2 and f2i+1,2 along this line (the difference of their ǫ ǫ u-coordinates) is doubled: δu(f12 ) = δu(f11 ) + µ(u0, v0 + ǫ)ǫ3 + o(ǫ3 ) = 2µ(u0 , v0 )ǫ3 + 3 ǫ ǫ 4 o(ǫ ), δv(f12 ) = −δv(f11 ) + ν(u0 , v0 + ǫ)ǫ + o(ǫ4 ) = o(ǫ4 ), Here and afterwards we use the following notations: δv(fijǫ ) ≡ v(fijǫ ) − v(fij ), δu(fijǫ ) ≡ u(fijǫ ) − u(fij ), where u(P ) and v(P ) denote the respective curvilinear coordinates u, v for any point P on the surface f (Ω). ǫ ǫ For the third row we will have δv(f13 ) = 3µ(u0 , v0 )ǫ3 +o(ǫ3 ), δu(f13 ) = ν(u0 , v0 )ǫ4 + o(ǫ4 ). ǫ This linear ”u-drift” of the points f1,k in contrast to ”v-oscillation” of the same points is easily explained: the position of the points fi0ǫ = fi0 = f (u0 + iǫ, v0 ) on the initial conjugate line v = v0 may be changed using a reparametrization u 7−→ u¯ = f (u) of the first conjugate coordinate u; thus the u-coordinates of the points fi1ǫ of the first row may be considered as some new choice of u-parametrization; all subsequent rows simply shift accordingly.

Lemma 10 For two adjacent infinitesimal curvilinear coordinate quadrangles ABDC and BDF E of ǫ-size (see Fig. 6) on a smooth conjugate net f : Ω −→ R3 without parabolic points one can find in a ǫ3 -neighborhood of D a unique point M on f (Ω) such that the quadrangles ABMC and BMF E are planar. The curvilinear coordinates of M are u(M) = u0 + ǫ + δuM , v(M) = v0 + ǫ + δvM , with δuM ∼ µ(u0, v0 )ǫ3 , δvM ∼ ν(u0 , v0 )ǫ4 . 15

Proof. We will follow the guidelines of the proof of Theorem 5. Using the Taylor expansions up to order O(ǫ4 ) for the Cartesian coordinates of A(u0 , v0 ), B(u0 +ǫ, v0 ), C(u0 , v0 + ǫ), D(u0 + ǫ, v0 + ǫ), E(u0 + 2ǫ, v0 ), F (u0 + 2ǫ, v0 + ǫ) and some point D1 (u0 + ǫ + δu1, v0 + ǫ + δv1 ) with some δu1 = O(ǫ3 ), δv1 = O(ǫ3 ) and taking −→ −→ −−→ into consideration (15), we compute the triple products W1 = ((AB × AC) · AD1 ) =   −−→ −−→ − −→ 3 ǫ6 γ(u0 , v0 )+ ǫ2 δu1 (fuu ×fu )·fv +δv1 (fvv ×fu )·fv +o(ǫ6 ), W2 = ((BE×BD1 )·F B) =   ǫ3 6 ǫ γ(u0 , v0 ) + 2 δu1 (fuu × fu ) · fv − δv1 (fvv × fu ) · fv + o(ǫ6 ) where γ(u, v) is a combination of the coefficients f (u, v), β(u, v) of (15), their derivatives and the triple products (fuu × fu ) · fv , (fvv × fu ) · fv . So if both of the latter triple products do not vanish, one can choose unique δu1 = µ ¯(u0, v0 )ǫ3 , δv1 = 0 such that W1 = O(ǫ7 ), W2 = O(ǫ7 ). This shows (as in the proof of Theorem 6) that D1 lies in ǫ4 - neighborhood of the curves IABC , IBEF of intersection of f (Ω) with the planes (ABC), (BEF ). Since the angle of intersection of IABC , IBEF is ≃ 1, we conclude that the point M of their intersection, close to D1 , is in fact ǫ4 -close to D1 . Using Taylor expansions for the same points up to order O(ǫ5 ), one can prove that for a generic surface actually δuM ∼ µ(u0, v0 )ǫ3 , δvM ∼ ν(u0 , v0 )ǫ4 with µ 6≡ 0, ν 6≡ 0. 2 D = f11

C = f01

F = f21

D1 M

A = f00

B = f10

E = f20

Figure 6: The adjacent infinitesimal smooth quads are defined by the points A, B, C, D, E and F (schematically shown as rectangles) and the planar quads ABMC, BMF E If one repeats the same calculation using instead of B a point B1 (u0 + ǫ + δu(B1 ), v0 + δv(B1 )) with some δu(B1 ) = O(ǫ2 ), δv(B1 ) = O(ǫ3 ), one will get from the conditions W1 = O(ǫ7 ), W2 = O(ǫ7 ) that δu1 = δu(B1 ) + µ(u0, v0 )ǫ3 + o(ǫ3 ), δv1 = −δv(B1 ) + ν(u0 , v0 )ǫ4 + λ(u0 , v0 )δv(B1 )ǫ + γ(u0 , v0 )δu(B1 )ǫ2 + o(ǫ4 ) where µ(u0, v0 ), ν(u0 , v0 ), λ(u0 , v0 ), γ(u0 , v0 ) are some algebraic combinations of the coefficients f (u, v), β(u, v) of (15), their derivatives and the triple products (fuu × fu ) · fv , 16

(fvv × fu ) · fv . Hence the following estimates hold for the second point M of intersection of IAB1 C and IB1 EF : Lemma 11 For two adjacent infinitesimal curvilinear coordinate quadrangles ABDC and BDF E of ǫ-size on a smooth conjugate net f : Ω −→ R3 without parabolic points and some point B1 (u0 + ǫ + δu(B1 ), v0 + δv(B1 )), δu(B1 ) = O(ǫ2 ), δv(B1 ) = O(ǫ3 ), one can find in a ǫ2 -neighborhood of D a unique point M on f (Ω) such that the quadrangles AB1 MC and B1 MF E are planar. The curvilinear coordinates of M are u(M) = u0 + ǫ + δuM , v(M) = v0 + ǫ + δvM , with δuM = δu(B1 ) + µ(u0, v0 )ǫ3 + o(ǫ3 ), δvM = −δv(B1 ) + ν(u0 , v0 )ǫ4 + λ(u0 , v0 )δv(B1 )ǫ + γ(u0 , v0 )δu(B1 )ǫ2 + o(ǫ4 ). Remark. As before, one can explain the “ǫ2 -tolerance” along the u-lines simply by removing this u-shift using a reparametrization of the curvilinear coordinate u. Note that we need this reparametrization only locally, for a given pair of elementary infinitesimal quadrangles, in order to establish this estimate for them, and not globally, for the complete discrete conjugate net. For global estimates we need the following simple result ([3]): Lemma 12 (Discrete Gr¨onwall estimate) Assume that a nonnegative function ∆ : N −→ R satisfies ∆(n + 1) ≤ (1 + ǫK)∆(n) + κ (21) with nonnegative constants K, κ. Then ∆(n) ≤ (∆(0) + nκ) exp(Knǫ).

(22)

Theorem 13 For a smooth conjugate net without parabolic points f : Ω → R3 and sufficiently small ǫ > 0, there exists a discrete conjugate net f ǫ with all its points fijǫ on f (Ω), such that d(fij , fijǫ ) = O(ǫ2 ). Proof. From Lemma 11 we immediately see that for each column k = 2i+1 the funcǫ tion ∆(n) = |δu(fkn )| satisfies the estimate (21) with K = 0, κ = 2ǫ3 max(u,v)∈Ω |µ(u, v)| for sufficiently small ǫ, so for all n we have (22). Since the number of steps in each ǫ column is ≃ 1/ǫ we obtain the global estimate |δu(f2i+1,j )| ≤ Cǫ2 . ǫ Now we set ∆(n) = |δv(fkn )| for each column k = 2i + 1. Taking the estimates ǫ of Lemma 11 and the already obtained global estimate for |δu(f2i+1,j )| we get (21) 4 with K = max(u,v)∈Ω |λ(u, v)|, κ = 2ǫ max(u,v)∈Ω (|µ(u, v)| + |γ(u, v)|), so (22) gives ǫ the global estimate |δv(f2i+1,j )| ≤ Cǫ3 . 2 ǫ Remark. As we have observed above, the v-shift of the points f2i+1,j has a much more interesting oscillating behavior; in particular one may conjecture that ǫ |δ(f2i+1,j )| = O(ǫ4 ) globally.

17

4.2

Curvature lines and circular nets

We start the construction of a circular discrete approximation of a given finite piece f (Ω) of a smooth surface without umbilic points parametrized by curvature lines fixing points fi0ǫ ≡ fi0 = f (u0 + iǫ, v0 ), f0iǫ ≡ f0i = f (u0 , v0 + iǫ) on two initial curvature lines passing throw a point f00 = f (u0 , v0 ) (cf. Figure 7).

ǫ f13

f03 =

ǫ f03

f13 ǫ f22

ǫ f02 = f02

ǫ f12

f12

ǫ f01 = f01

ǫ f11

f11

f22

ǫ f21

f31 f21

ǫ f31

ǫ f30 = f30

f00 =

ǫ f00 ǫ f10 = f10

ǫ f20 = f20

Figure 7: A circular net fijǫ with its vertices on f (Ω), which are ǫ2 -close to the vertices of the smooth net of conjugate lines fij The first circle is defined by the triple {f00 , f10 , f01 }, its fourth point of intersecǫ tion with the surface will be chosen as the next constructed point f11 of the discrete circular net. On the next step we can define two circles by the triples of points ǫ ǫ {f01 , f02 , f11 } and {f10 , f20 , f11 }, the fourth points of intersection of these circles ǫ ǫ and f21 . The behavior of the new points on the with the surface will give us f12 third step is of crucial importance. Their error estimates can be obtained from the following generalization of Theorem 7: Lemma 14 Let the points A, B and C on f (Ω) be chosen, such that u(A) = u0 + δuA , v(A) = v0 + δvA , u(B) = u0 + ǫ + δuB , v(B) = v0 + δvB , u(C) = u0 + δuC , v(C) = v0 +ǫ+δvC , with all δu, δv of order = O(ǫ3 ). Then the curvilinear coordinates of the fourth point M of intersection of the circle ωABC with f (Ω) are: u(M) = u(f11 ) + δuM , v(M) = v(f11 ) + δvM , with δuM = δuA + δuB − δuC + ǫ(δvB − δvA )µ1 (u0 , v0 ) +ǫ(δuC − δuA )µ2 (u0, v0 ) + µ3 (u0 , v0 )ǫ3 + µ4 (u0 , v0 )ǫ4 + o(ǫ4 ), δvM = δvA − δvB + δvC + ǫ(δvB − δvA )ν1 (u0 , v0 ) +ǫ(δuC − δuA )ν2 (u0 , v0 ) + ν3 (u0 , v0 )ǫ3 + ν4 (u0 , v0 )ǫ4 + o(ǫ4 ), 18

(23)

Proof. We will use the same quaternionic cross-ratio Q = (A − B)(B − C)−1 (C − D)(D − A)−1 . The obvious changes are necessary in the Taylor expansions (19), where we shall take into consideration the δu- and δv-shifts of the points; the expansions themselves should include terms up to order O(ǫ4 ). For simplicity we will assume δuA = 0, δvA = 0, the general case requires only a shift by δuA , δvA in the net of curvature lines on f (Ω). Fixing for the moment the points A, B, C and introducing D ∈ f (Ω), u(D) = u0 + ǫ + δuD , v(D) = v0 + ǫ + δvD with some δuD = O(ǫ3 ), δvD = O(ǫ3 ), one can compute the imaginary part of Q:
Im(Q) = I ρ1 (u0 , v0 )ǫ3 + θH2 (δvD − δvC + δvB ) + o(ǫ3 )
+J ρ2 (u0 , v0 )ǫ3 + θH1 (−δuD − δuC + δuB ) + o(ǫ3 )
+K ρ3 ǫ2 + ǫ−1 H1 H2 (H12 (δuB − δuD − δuC ) + H22 (δvD + δvC − δvB )) + o(ǫ2 ) , where θ = 21 (H12 H2 β32 − H1 H22 β31 ) = H12 H22 (K2 − K1 )/2, Ki being the principal curvatures at the point A. Since we assume K1 6= K2 , choosing δuD = δuB − δuC + ǫ3 µ3 (u0 , v0 ) + o(ǫ3 ), δvD = −δvB + δvC + ǫ3 ν3 (u0 , v0 ) + o(ǫ3 ), one will achieve Im(Q) = o(ǫ3 )I + o(ǫ3 )J + o(ǫ2 )K. As a lengthier computation shows, adding appropriate fourth-order corrections in Taylor expansions, given in (23) (set there δuA = 0, δvA = 0), we get Im(Q) = o(ǫ4 )I + o(ǫ4 )J + o(ǫ3 )K. This means that the point D with δuD , δvD given in (23) is ǫ6 -close to the plane (ABC) −→ −→ −−→ (this can be also checked using the triple product (AB × AC) · AD) and ǫ5 -close to the circle ωABC passing through A, B, C. Using the the fact that the angle between f (Ω) and the plane (ABC) is ≃ ǫ at D, we conclude that for the fourth point M of intersection of ωABC and f (Ω) we can keep the same expansions (23). 2 Remark. Linear behavior of δuD and δvD in (23) is valid precisely in orders O(ǫ3 ) and O(ǫ4 ), corrections of order O(ǫ5 ) are nonlinear w.r.t. δu- and δv-shifts of the points A, B and C. ǫ ǫ ) − δu(f02 ) + µ3 (u0 , v0 + ǫ)ǫ3 + Using (23) we calculate δu(f12 ) = δu(f01 ) + δu(f11 ǫ ǫ ǫ o(ǫ3 ) = 2µ3 (u0 , v0 )ǫ3 + o(ǫ3 ), δv(f12 ) = δv( f01 ) − δv(f11 ) + δv(f02 ) + ǫ(δv(f11 )− 3 δv(f01 ))ν1 (u0 , v0 + ǫ) + ǫ(δu(f02 ) − δu(f01 ))ν2 (u0 , v0 + ǫ) + ν3 (u0 , v0 + ǫ)ǫ + ν4 (u0 , v0 + ǫ)ǫ4 + o(ǫ4 ) = ǫ4 (∂v ν3 (u0, v0 ) + ν1 (u0 , v0 )ν3 (u0 , v0 )) + o(ǫ4 ). As we see now, for ǫ ǫ points f12 (respectively f21 ) only the shifts along the respective curvature lines is 3 of order O(ǫ ) (only for surfaces of special type this shift may degenerate to 0), ǫ while the perpendicular shift is of order O(ǫ4 ). For f22 we now obtain a remarkable 4 result: both its shifts are of order O(ǫ ), compared to the error estimates of order ǫ ǫ ǫ ) = 2ǫ4 (∂u µ3 (u0, v0 ) + µ2 (u0 , v0 )µ3 (u0 , v0 )) + o(ǫ4 ), δv(f22 ) = : δu(f22 O(ǫ3 ) for f11 4 4 2ǫ (∂v ν3 (u0 , v0 ) + ν1 (u0 , v0 )ν3 (u0 , v0 )) + o(ǫ ). 19

This observation suggests us to partition the complete discrete lattice fijǫ into 3 sublattices: ǫ a) the even sublattice of points f2i,2j , ǫ b) the odd sublattice of points f2i+1,2j+1 , ǫ ǫ c) the intermediate sublattice of points f2i+1,2j , f2i,2j+1 . ǫ Infinitesimally, in a (Nǫ)-neighborhood of the initial point f00 (with N ≪ 1/ǫ), 4 the even sublattice has both δuD - and δvD -shifts of order O(ǫ ), for the odd sublattice they have order O(ǫ3 ), for the intermediate one only the shifts along the respective curvature lines are of order O(ǫ3 ), the perpendicular shifts being of order O(ǫ4 ) again. Accumulation of errors for these sublattices is also different: linear accumulation of O(ǫ3 )-errors for the odd sublattice, quadratic for the even sublattice and mixed for the intermediate sublattice: linear accumulation of O(ǫ3 )-shifts along the respecǫ tive curvature lines and quadratic in perpendicular direction: δu(f2i,2j ) ≃ i · j · ǫ4 , ǫ 4 ǫ 3 ǫ δv(f2i,2j ) ≃ i · j · ǫ , δu(f2i+1,2j+1 ) ≃ (i + j) · ǫ , δv(f2i+1,2j+1 ) ≃ (i + j) · ǫ3 , ǫ ǫ δu(f2i,2j+1 ) ≃ i · j · ǫ4 , δv(f2i,2j+1 ) ≃ (i + j) · ǫ3 . These estimates are easily obtained using (23) (see below the proof of Theorem 15), for example the first one follows from: ǫ ǫ ǫ ǫ δu(f2(i+1),2(j+1) ) = δu(f2(i+1),2j ) + δu(f2i,2(j+1) ) − δu(f2i,2j )+

ǫ ǫ )) + o(ǫ4 ). 2ǫ4 (∂u µ3 + µ2 µ3 ) − 2ǫµ2 (δu(f2i,2(j+1) ) − δu(f2i,2j

(24)

Note that the estimates of Lemma 14 do not require the initial three points A, B, C to lie near the grid points fij of the original smooth net and the distances d(A, B), d(A, C) need not to be equal, both just have to be of order O(ǫ). Theorem 15 For a smooth surface without umbilic points parametrized by curvature lines f : Ω → R3 , Ω = {(i, v)|u2 + v 2 < 1} and sufficiently small ǫ > 0, there exists a discrete circular net f ǫ with all its points fijǫ on f (Ω), such that d(fij , fijǫ ) = O(ǫ2 ). ǫ Proof. We will give the details for the global estimate δu(f2i,2j ) ≤ C · i · j · ǫ4 , the other are proved in the same way. ǫ ǫ ) − δu(f2k,2n )|. From (24) we see First define a function S(k, n) = |δu(f2k,2(n+1) that S(k + 1, n) ≤ S(k, n) + 2ǫ|µ2 |S(k, n) + 2ǫ4 |∂u µ3 + µ2 µ3 | + o(ǫ4 )

and using Lemma 12 we immediately obtain S(k, n) ≤ Cǫ4 k. ǫ Now for the functions ∆k (n) = |δu(f2k,2n )| taken separately for each column of the even sublattice one gets from (24) that

∆k (n + 1) ≤ ∆k (n) + (1 + 2ǫ|µ2 |)S(k, n) + 2ǫ4 K ≤ ∆k (n) + ǫ4 (2Ck + K) 20

with K = 2 max(u,v)∈Ω |∂u µ3 + µ2 µ3 |. So (22) gives us the global estimate ∆k (n) ≤ C¯ · k · n · ǫ4 2 In [3] one can find a similar result with the same order ǫ2 of approximation but without the requirement fijǫ ∈ f (Ω).

Acknowledgments The authors wish to thank Dr. I. Dynnikov for the idea of simplification of the proof of Lemma 2 and Prof. P. Schr¨oder for valuable discussions.

Appendix Lemma 16 Let an ǫ-neighborhood f : Ωǫ → R3 of a point M on a smooth surface be given, and for the principal curvatures one has K12 + K22 > 0 at M. Then the intersection IABC of a plane πABC lying at ǫ2 -distance from the tangent plane πM to f (Ωǫ ) at M is a curve lying in ǫ2 -neighborhood of a quadric — the corresponding Dupin indicatrix of the surface f (Ω) in the central point M of the ǫ-patch. Proof. Taking the appropriate Cartesian coordinate system with the origin M we can approximate the chosen ǫ-patch by the osculating paraboloid P = {z = (K1 x2 + K2 y 2 )/2} with error terms of order o(ǫ2 ). We obtain the quadric QM = P ∩ πABC called the Dupin indicatrix of f (Ω) at M. As one can easily check, the angle between πM and any of the tangent planes, taken at a point of f (Ωǫ ), ǫ2 -close to QM , is ≃ ǫ: the normals ~n = fu × fv to f (Ωǫ ) at such points are ǫ2 -close to the normals ~n1 of the osculating paraboloid at the corresponding points with the same (x, y)-coordinates. For the latter one has ~n1 (x, y) = −K1 x~i + K2 y~j + ~k and for x ≃ ǫ and/or y ≃ ǫ, the angle between ~n1 (x, y) and ~n1 (0, 0) = ~k is ≃ ǫ. Now using the fact that f (Ωǫ ) is ǫ3 -close to the osculating paraboloid and standard estimates for the values of implicit functions and their derivatives we conclude that IABC and QM are ǫ2 -close and the tangent directions at their ǫ2 -close points are also ǫ2 -close. 2

References [1] P. Alliez, D. Cohen-Steiner, O. Devillers, B. L´evy,, m. Desbrun, Anisotropic polygonal remeshing. ACM Transactions on Graphics. v. 22 , Iss. 3 (2003) p. 485–493. [2] A.I. Bobenko, Yu.B. Suris, Discrete differential geometry. Consistency as integrability, Preliminary version of a book (2005). Preprint arXiv:math.DG/0504358.

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[3] A.I. Bobenko, D. Matthes, Yu.B. Suris, Discrete and smooth orthogonal systems: C ∞ -approximation, Internat. Math. Research Notices 45 (2003), 2415– 2459. Also preprint arXiv:math.DG/0303333. [4] A.I. Bobenko, U. Pinkall. Discrete isothermic surfaces. J. reine andew. Math. 1996 v. 475, p. 187–208. [5] A.I. Bobenko, Yu.B. Suris, On organizing principles of Discrete Differential Geometry. Geometry of spheres. preprint arXiv:math.DG/0608291, 2006. [6] G. Darboux. Le¸cons sur la th´eorie g´en´erale des surfaces et les applications g´eom´etriques du calcul infinit´esimal. T.I–IV. 3rd edition. Paris: GauthierVillars, 1914–1927. [7] G. Darboux, Le¸cons sur les syst`emes orthogonaux et les coordonn´ees curvilignes, Paris (1910). [8] J. Goldfeather, V. Interrante, A novel cubic-order algorithm for approximating principal direction vectors. ACM Transactions on Graphics. v. 23 , Iss. 1 (2004) p. 45–63. [9] U. Hertrich-Jeromin, Introduction to Mobius differential geometry. Cambridge University Press, 2003. xii+413 pp. [10] J. Milnor, Morse theory. Princeton, NJ: Princeton Univ. Pr., 1970. - VIII, 153 p. (Annals of mathematics studies; v. 51) [11] D. Matthes. Discrete surfaces and coordinate systems: approximation theorems and computation. PhD TU-Berlin, 2003, http://edocs.tu-berlin.de/diss/2003/matthes daniel.htm [12] H. Pottmann, Y. Liu, J. Wallner, A. Bobenko, W. Wang, Geometry of Multilayer Freeform Structures for Architecture, ACM Trans. Graphics 26(3) (2007), SIGGRAPH 2007.

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